Beam Deconvolution Map-Making

Overview

We present a map-making method for data from scanning CMB telescopes. The method implements fast algorithms for convolution and transpose convolution of two functions on the sphere. We allow for arbitrary beam asymmetries and scanning strategies. The algorithm succeeds in removing artifacts due to these beam asymmetries and far sidelobes and outperforms the standard map-making method in quantitative comparison.

Recent Noisey Results

[noise power spectrum overplotted the true sky power spectrum]
Figure 1

Plotted here is the power spectrum of the true sky (in black). Overplotted in green and red are the power spectra of the noise that we add to the maps. The red line represents the noise added to the WMAP-like scan (WSP) maps and the green line represents the noise added to the basic-scan (BSP) maps. The purpose of this plot is to demonstrate that noise dominates at high ell.

The least squares estimate of the true sky, s, is given by ATAs = ATd. The coefficient matrix, ATA, is a smoothing matrix and therefore ill-conditioned. We introduce a regularization technique by factoring the convolution operator A into A = BG, where G is a simple Gaussian smoothing matrix and is parametrized by its FWHM value.


[noiseless vs. noisey results with 180 regularization]
Figure 2

Ratios of the spectra of the residual maps to the spectrum of the input map are displayed in Figure 2. The noise-less results are plotted in black and the noisey results are in red. Solid lines correspond to the deconvolved spectra and the dashed lines correspond to the standard spectra. The sidelobe beam is composed of a Gaussian beam of FWHM=1800' rotated at 90 degrees to another Gaussian beam of FWHM=180'. The elliptical beam is composed of two Gaussian beams with FWHM=180' whose centers are separated by 180'.
Note that the residual Cl's of the noisey maps are divided by the power spectrum of the input map plus the noise, while the residual Cl's of the noise-less maps are only divided by the power spectrum of the input map. Unless otherwise stated, the FWHM of the regularization is 180'.


[ratios of spectra of residual maps]
Figure 3

This is the same plot as above but excludes the noise-less results.
We see, in the sidelobe case, that we recover the true sky to greater accuracy using the deconvolution method at low ell. The two methods give similar results at high ell, where the noise dominates (recall Figure 1). In the elliptical beam case, the attempt to recover resolution where the noise dominates leads to an amplification of the noise.


[noise180 vs. noisey225 for elliptical beam case]
Figure 4

In this plot we examine just the elliptical beam case. The residual spectra plotted in black are the same as above (i.e., FWHM of 180'). In red, we have changed the regularization parameter to a FWHM of 225'. We see that the amplification of the noise at high ell seen in Figure 3 is suppressed by reducing the target resolution. This is done by increasing the regularization scale from 180' to 225'.